Problem 38 - Precalculus: Functions and Graphs

Question

If one zero of \(f(x)=x^{3}-3 x^{2}-k x+12\) is \(-2,\) find two other zeros.

Step-by-Step Solution

Answer

The other zeros of the polynomial are 2 and 3.

Step 1: Verify the Given Zero

We need to substitute \(x = -2\) into the polynomial \(f(x) = x^3 - 3x^2 - kx + 12\) and ensure that it satisfies \(f(x) = 0\). If \(-2\) is indeed a zero, the equation should hold true.

Step 2: Substitute and Simplify

Substitute \(x = -2\) into the polynomial:\(-2^3 - 3(-2)^2 - k(-2) + 12 = 0\)This simplifies to:\(-8 - 12 + 2k + 12 = 0\)Simplifying further, \(2k - 8 = 0\).

Step 3: Solve for k

Solve the equation for \(k\):\[2k - 8 = 0\] which leads to:\[2k = 8\]\[k = 4\]

Step 4: Factor the Polynomial

With \(k = 4\), the polynomial becomes \(f(x) = x^3 - 3x^2 - 4x + 12\). Use synthetic division with \(x + 2\) to find other factors:-2 | 1 -3 -4 12 | -2 10 -12 ---------------- 1 -5 6 0The quotient is \(x^2 - 5x + 6\).

Step 5: Find Remaining Zeros

Factor the quadratic \(x^2 - 5x + 6\):\(x^2 - 5x + 6 = (x - 2)(x - 3)\).Thus, the remaining zeros are \(x = 2\) and \(x = 3\).

Key Concepts

synthetic division

When exploring polynomial functions, synthetic division is an efficient method for dividing a polynomial by a simple binomial of the form \(x - c\). This technique is especially useful when you know one zero of a polynomial, as it helps divide the polynomial to find other zeros or factors.
To use synthetic division, follow these steps:

• Write down only the coefficients of the polynomial in descending order of degree.
• Identify the zero or root \(c\). In this example, it is \(-2\).
• Write \(c\) on the left and the coefficients horizontally. Apply the division algorithm using addition and multiplication to fill in the new row below.

At the end of the division, the final row will represent the coefficients of the quotient polynomial, while the remainder should be zero if \(c\) is truly a zero.
This method simplifies calculations and computation, reducing errors commonly made in long polynomial division.

factoring quadratics

Factoring quadratics is a way of finding the zeros of a quadratic polynomial, which is a polynomial of degree 2. Once you have factored the polynomial, you can set each factor equal to zero to find the solutions or roots.
The basic form of a quadratic polynomial is \(ax^2 + bx + c\). To factor it, you need to find two numbers that multiply to give \(c\) (the constant term) and add up to \(b\) (the coefficient of \(x\)).
For the quadratic \(x^2 - 5x + 6\), finding these numbers is straightforward:

• The factors of 6 that add up to -5 are -2 and -3.
• So this quadratic can be expressed as \((x - 2)(x - 3)\).

This factorization makes it easy to discover that the zeros of the quadratic are \(x = 2\) and \(x = 3\).
Factoring quadratics is a powerful tool for handling polynomials encountered in algebra problems.

polynomial functions

Polynomial functions are mathematical expressions involving a sum of powers of \(x\). Each term in a polynomial is made up of a coefficient and a variable raised to a non-negative integer exponent. The degree of the polynomial is determined by the highest power of \(x\) present.
For example, the function \(f(x) = x^3 - 3x^2 - 4x + 12\) is a third-degree polynomial because the highest power of \(x\) is 3.
Polynomials can have various characteristics, like:

• The number of zeros, which are determined by its degree. A third-degree polynomial can have up to three real zeros.
• Symmetry, direction of opening (up or down), and end behavior based on the leading coefficient and the degree.

Understanding the nature of these functions is crucial for graphing and solving algebraic equations. Key operations in polynomials include addition, subtraction, multiplication, division, and factoring.

zero verification

To verify a zero of a polynomial, substitute the alleged zero into the polynomial and simplify. If the result is zero, the substitution confirms that the number is indeed a valid zero or root.
In the given problem, substituting \(x = -2\) into \(f(x) = x^3 - 3x^2 - 4x + 12\) helps ascertain its zero status:
Upon substituting, we had:
\(-2^3 - 3(-2)^2 - 4(-2) + 12\), which simplifies to \(-8 - 12 + 8 + 12 = 0\).
The equation simplifies neatly to zero, thus verifying \(-2\) as an accurate zero.
This step is critical because it ensures accuracy before proceeding to further calculations such as synthetic division or factoring. Zero verification should always be the first step when dealing with potential zeros of polynomials.